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WEINGARTEN SURFACES ARISING FROM SOLITON THEORY
A THESIS
SUBMITTED TO THE DEPARTMENT OF MATHEMATICS
AND THE INSTITUTE OF ENGINEERING AND SCIENCES
OF BILKENT UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF MASTER OF SCIENCE
ByÖzgür Ceyhan August, 1999
I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science.
; ( ^ ¿ i .Prof. Dr. Metin Gürses(Principal Advisor)
I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science.
Prof. Dr. Alexander Klyachko
I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science.
/
Prof. Dr. Turgut Önder
Approved for the Institute of Engineering and Sciences:
Prof. Dr. Mehmet Director of Institute of Engineering and Sciences
11
ABSTRACT
WEINGARTEN SURFACES ARISING FROM SOLITON THEORY
Özgür Ceyhan M. S. in Mathematics
Advisor: Prof. Dr. Metin Gürses August, 1999
In this work we presented a method for constructing surfaces in associated with the symmetries of Gauss-Mainardi-Codazzi equations. We show that among these surfaces the sphere has a unique role. Under constant gauge transformations all integrable equations are mapped to a sphere. Furthermore we prove that all compact surfaces generated by symmetries of the sine-Gordon equation are homeomorphic to sphere. We also construct some Weingarten surfaces arising from the deformations of sine-Gordon, sinh-Gordon, nonlinear Schrödinger and modified Korteweg-de Vries equations.
Keywords and Phrases: Solitons, integrable surfaces, Weingarten surfaces.
Ill
ÖZET
SOLİTON TEORİSİNDEN TÜRETİLEN WEINGARTEN YÜZEYLERİ
Özgür CeyhanMatematik Bölümü Yüksek Lisans Danışman: Prof. Dr. Metin Gürses
Ağustos, 1999
Bu çalışmada Gauss-Mainardi-Codazzi denklemlerinin simetrileri ile bağıntılı teki yüzeyleri türetmek için bir yöntem verildi. Bu yüzeyler arasında kürenin özel bir yeri olduğu belirlendi. Tüm entegre edilebilir denklemlerin sabit ayar dönüşümlerinden elde edilen yüzeylerin küre olduğu kanıtlandı. Ayrıca sine-Gordon denkleminin simetrileri kullanılarak türetilen tüm kompakt yüzeylerin küreye homeomorfik olduğu gösterildi. Sine-Gordon, sinh-Gordon, doğrusal olmayan Schrödinger ve değişik Korteweg-de Vries denklemlerinin simetrileri ile bağıntılı bazi Weingarten yüzeyleri verildi.
Anahtar Kelimeler ve ifadeler: Solitonlar, entegre edilebilir yüzeyler, Weingarten yüzeyleri.
İV
to Berna
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to Prof. Dr. Metin Gürses for his excellent supervision. His patience and expert guidence bring my research up to this point.
I am greatful to Burak Gürel who read large portions of manuscript at an early stage, and his detailed comments enhanced both the content and the style of the final product.
I owe special thanks to my family, in particular Metin Eşmen whose en- couregement have supported my graduate study from its begining.
I am indebted to all my friends for all they have done for me.
As on several previous occasions, the most thanks goes to my best friend who offered encouragement, understanding, patience, and help in many different ways. Thanks, Nur.
VI
Table of Contents
1 In troduction 1
2 Surfaces in 3
2.1 Elements of the Theory of S u rfaces................................................ 3
2.2 Gauss and Mainardi-Codazzi Equations.......................................... 4
3 Soliton Surfaces 7
3.1 Surfaces Immersed in E^ as Surfaces in Lie Algebras.................... 7
3.1.1 Immersions in E ^ ................................................................... 8
3.1.2 Soliton Surfaces A pp roach ................................................... 9
3.1.3 The Fokas-Gelfand Approach 10
3.2 A Generalized Immersion F u n ction ................................................ 13
3.3 Immersions Associated With The Symmetries of The IntegrableGauss-Mainardi-Codazzi E quations................................................ 17
3.3.1 Immersions Associated With the Constant Gauge Transformations ................................................................................ 18
3.3.2 Immersions Associated With The Sine-Gordon Equation 19
4 W eingarten Surfaces 25
vii
4.1 Linear Weingarten Surfaces 25
4.1.1 The Sine-Gordon Equation................................................... 25
4.1.2 The Sinh-Gordon Equation 26
4.2 Nonlinear Weingarten Surfaces......................................... 29
4.2.1 The Nonlinear Schrôdinger E qu ation ................................ 29
4.2.2 The mKdV E qu ation ............................................................. 30
5 C onclusion 34
A Fundam ental Equations for Subm anifolds 35
A .l Fundamental Equations for Submanifolds....................................... 35
A .2 Gauss, Mainardi-Codazzi and Ricci Equations in Local Coordinates ....................................................................................................... 38
A.3 Immersions into Constant Curvature S p a ce s ................................ 40
A .4 Immersions of Hypersurfaces........................................................... 41
vin
Chapter 1
Introduction
The latter period of the nineteenth century and the early part of this century saw a great deal of activity in the study of special classes of surfaces in three dimensional Euclidean space (see, e.g. [27]-[31]). Typical examples include minimal surfaces, surfaces of constant mean curvature and surfaces of constant Gaussian curvature. Gauss equations that describe surfaces in three dimensional space have been studied in detail from various points of view. One of the classical problems of differential geometry was the study of the connections between geometry of submanifolds and nonlinear partial differential equations (PDEs). Probably sine-Gordon and Liouville equations are the best known examples. They describe minimal and pseudospherical surfaces respectively. They arise as the compatibility condition of the Gauss-Weingarten equations of a surface under a suitable parametrization. At that time many features of integrability of the sine-Gordon, Liouville and some other integrable equations were discovered.
On the other hand, the works of Kruskal-Zabursky, Lax, AKNS, Zakharov- Shabat,... introduced a technique (inverse spectral transform) for solving nonlinear PDEs, in the 1960’s (see, e.g. [21]-[25]). This method allows one to solve a number of nonlinear PDEs. Nonlinear PDEs integrable by the inverse spectral transformation possess some remarkable properties such as soli ton solutions, an infinite number of conservation laws, infinite symmetry groups, special Hamiltonian structures,...
A key element of the inverse spectral transformation method is the representation (Lax representation) of the nonlinear PDE
Ul,2-U2,l + [U i,U 2]=0
as a compatibility condition of certain system of linear equations
= k = l,2 .
Lax representation has a transparent geometrical interpretation. We may identify these equations with Gauss-Mainardi-Codazzi (GMC) equations represented as the compatibility condition of linear equations for the moving frame (Gauss-Weingarten equations). Due to the analogy between GMC and Lax equations, for a long time, surface theory was used as a source of integrable equations (see e.g [8]-[20]). In the last decade, the attitude is to use soliton theory in understanding some local and global properties of surfaces, (e.g. [1]-[14]).
In this work we investigate the relationship between the generalized symmetries and the associated surfaces in R . In chapter 2, readers are reminded of the basic notions and equations of differential geometry of surfaces. Sym’s formulation of soliton surfaces and its recent generalization given by Fokas and Gelfand are presented in the first section of chapter 3. In the next section, a general method of constructing immersion functions by using the symmetries of GMC equations is discussed. In following sections of chapter 3, some local and global properties of particular surfaces are described. We show that surface associated with constant gauge transformation is a sphere and investigate the symmetry surfaces of the sine-Gordon equation. We show that compact, connected, oriented sine-Gordon surfaces are homeomorphic to sphere. In last chapter, we constructed several Weingarten surfaces arising from the symmetries of the sine-Gordon, sinh-Gordon, nonlinear Schröndinger and modified Korteweg-de Vries equations. In Appendix, Gauss-Mainardi-Codazzi-Ricci equations are given for higher dimensional embedded or immersed manifolds of arbitrary codimensioris.
Chapter 2
Surfaces in M?
In this chapter we shall give a brief survey of two dimensional surfaces immersed in For the Gauss-Mainardi-Codazzi equations, we use the corresponding equations given in Appendix D for dimension m = 2. For further details of two dimensional surfaces see [33, 34].
2.1 Elements of the Theory of Surfaces
Our interest is now directed toward some elementary concepts of surfaces immersed in
D efin ition 2.1 Let M C R^ a surface, with the inclusion map F : U C R" —> R^ . Then the first fundamental form (or equivalently induced metric g) is F* > where < .,. > is the usual inner product on R^.
We write the first fundamental form tensor on M as
{dsiY = 51 idx ® dx + 2gudx^ O dx -f g22dx ® dx^,
where we define functions gij, i , j = 1,2 directly by using the local coordinates {x\x^) of £/ C R^ with F : U ^ R ^
a·· - < ^ >^ 9a;· ’ dx3= < F ^ i ,F j> , i , j = 1,2.
In the sequel we shall use lower indices ” ,i” for the differentiation with respect to the coordinate x .
We next deal with the properties of the Gauss map u : M ^ 5” , namely a unit normal differentiable vector field, which can at least be defined in a neighborhood of each point p € M. (M is assumed to be orientable)
D efin ition 2.2 In terms of v, the second fundamental form H on M is defined as
n{p ) {X ,Y ) = < - d v { X ) , Y > ,
In particular, by considering an immersion F : U M. , for U C IR , the second fundamental form can be defined directly on U in terms of local coordinates by
bij <c t'jj·, F j > <c t', F ij > .
D efin ition 2.3 Let the matrix S with the coefficients 6*· = g' hkj represent the ’’shape operator” . Eigenvalues k\ and k,2 of S, are defined as the principle curvatures and then Gauss curvature K and mean curvature H are defined as
K = det[S) = k\k2 .,
H = tr{S) = kr + k2.
D efin ition 2.4 If there exists a function f such that f { K , I I ) — 0 (or equivalently f ( k i , k 2 ) = 0/ then the corresponding surface is called a Weingarten surface.
2.2 Gauss and Mainardi-Codazzi Equations
Let F : U C > M C be a local parametrization . Then it is possible to assign a trihedron to every point p ^ M given by the vectors F]i, F 2 and v.
We may express motion of this frame along M by the Gauss and Weingarten equations
F,ii = r* ./·,! + 6«, >/
Q , hji^j^k = 1,2,
where summation over repeated indices is assumed.
( 2 .1)
(2.2)
The above set of partial differential equations are integrable if certain compatibility conditions are satisfied. Setting F ijk = F ikj and assuming the linear independence of ^ 1,^2 and 1/, we get:
L em m a 2.5 Integrability conditions of (2.1) and (2.2) reduce to set of equations:
r'itj - r',., + r5.rl,. - r|r'j = bi,b· -
bih,j - bij k + = 0.(2.3)(2.4)
P roo f: m =2 case of theorem (A.3), (A .4) and lemma (A.6). □
Right hand side of the equation (2.3) is the Riemann curvature tensor. Then
R 12 1 2 = ¿11 *22 ■“ bi'ibu,
is the Gauss equation for surfaces in Hence that intrinsically defined Gauss curvature K is given by (see [32, 33]):
_ ___________ < R j F ^ l , F ^ 2)F ^ 2 ·, F ^ I > ___________ _ 611622 ~ ^12^12
< F . u F ^ i > < F^2- , F ,2 > — < F^ i , F 2 gn922 — 912912'
and if we take a look at equation (2.4), it reduces to the following set of equations
612,1 — 611,2 + r j2 6 ; i i — r i j 6 / i2 — 0 ,
622,1 — 621,2 + T226/11 — r 2 i 62 = 0 ,(2.5)
which are called Mainardi-Codazzi equations. The integrability conditions Uij = i/ji are satisfied automatically by the Mainardi-Codazzi equations.
E xam ple 1: (Surfaces of Revolution) Let M C be the set obtained by rotating a regular plane curve C about an axis in the plane which does not meet the curve; let the xz plane be the plane of curve the C and the z axis be
the rotation axis. Let C be parametrized by « (x ) = {(¡>{x),0,tp{x)). We can compute the ChristofFel symbols for a surface of revolution parametrized by:
F{x^,x^) = (<^(x^)cosX^,^(x^)sinx\ 5
Since
gu = ^(x'^Y , 5(12 == 0 , .922 = <f>{x‘ y +
we obtain the Christoifel symbols to be:
rli - 0 , r?, =
p i _ p2 _ Q 12 - ^2 ’ 12 -
pi _ n22 - ^ ’ 22 (^,)2 + (^.)2·
If we let x2 be the arclength parameter of the curve (i.e. {<f>'Y +then the Gauss and mean curvatures are given by
r r .
^ = ' 7 ’ " = 2 1,
and the Gauss equation (2.3) reduces to
(j)" + K(j) = 0.
Mainardi-Codazzi equations (2.5) are identically satisfied.
= 1),
Chapter 3
Soliton Surfaces
It is well a known fact that existence of Lax pair for a differential equation entails existence of infinitely many symmetries. The symmetry group of system of PDEs is the group of transformations that map solutions of the system to other solutions. Here in this chapter we shall present an explicit formulation of the immersion functions that associated with each symmetry of a given soliton equation.
3.1 Surfaces Immersed in M ? as Surfaces in Lie Algebras
Our goal in this section is to reformulate the classical theory of surfaces in a form familiar to the soliton theory, which makes an application of the analytical methods of this theory to integrable cases possible.
Formulas for the moving frames associated with integrable equations can be integrated. This issue was first suggested by A. Syrn [8]-[14] and generalized by Fokas and Gel’fand [2], Fokas, Gel’fand, Finkel, Liu [3] and Ciesliriski [6]. This approaches were applied to several soliton equations [1]-[14].
In section (3.1.1), the moving frame for a general surface is described in terms of s'u(2) algebra. In the sections (3.1.2) and (3.1.3), Sym, Fokas-Gel’fand, and Ciesliiiski approaches are presented.
3.1.1 Immersions in
Let F : U C Af C be an immersion and u{x^,x^) be the unit normalfield along M. Then F i, F 2 and u define a basis in As we have seen in previous chapter, the motion of this basis on M is characterized by Gauss- Weingarten equations (2.1), (2.2). Now let us consider following orthonormal basis
_ JP',! _ 9 u F 2 - g u F , i,--- J 2 —
\ / ^ yjgndet{g)
Let us consider this moving frame on M in 3 x 3 matrix form E' = (ci, C2, i')· Then the Gauss-Weingarten equations for the frame E become
E k = ^kE , fc = 1,2,
and Gauss-Mainardi-Codazzi equations are
Ai,2 — A2,i -b [Ai, A2] = 0,
(3.1)
(3.2)
where the matrices E and A^, = 1,2 have value in 5 '0(3) and so(3) respectively. It is convenient to use the isomorphism so(3) ~ su(2) to rewrite equation (3.1) in terms 2 x 2 complex matrices. Let ^{x^,x^) be an SU{2) valued function, then we can write these matrices explicitly as follows
where
$ , k = 1,2,
- t ) '
(3,3)
and
o k = 9nb2k — duhk
— — ' , Pk = ---------------------ib'Ik
9n \Jg\\det{g) V ?ir
We rewrite the compatibility conditions given by (3.2) as
C /l.2 -t /2 ,l-[t /l ,C /2 ]= 0 . (3.4)
Finally we summarize the result given in above arguments with following theorem.
8
T h eorem 3.1 [2] Let Uk = C4(x^,a;^) G su{2),k = 1,2 be differentiable functions of x^,x^ in some neighborhood o/M^. Assume that each Uk satisfy (3.4) then equations (3.3) define a 2-dimensional surface $ G SU{2).
R em ark 3.2 In the context of integrable systems equation (3.3) is known as the Lax equation and equation (3.4) o,s the zero curvature condition. However, in order to apply inverse, spectral transform one needs to insert a spectral parameter in (3.3), In the following sections we shall consider such cases.
3.1.2 Soliton Surfaces Approach
An interesting connection between classical geometry of surfaces and the symmetries of soliton equations is first given by Sym in [8]-[14].
T h eorem 3.3 [8] Let Uk = A) G su(2),k = 1,2 be differentiablefunctions of x^,x^ and A which satisfy (3.3) and (3.4). Assume that Gauss- Mainardi-Codazzi equations (3.4) are independent of X. Then
define a tangent space and
oX
(3.5)
(3.6)
defines an explicit immersion function of the surface associated with the A translation symmetry of equation the (3.4) where C is constant su{2) matrix.
P roo f: We define an SU{2) valued function by F{x,X) = $ ^(x, Ao)$(x, A) which is known as the Pohlmeyer transformation. The equation (3.3) yields
F,k = XoW kA x, A,)(A - Ao) + · · -]$(x, Ao)F (3.7)
whose integrability conditions of equation (3.7) (i.e. F ki = fiik) are
($->t/)t,A$),/ = ($-'C/,,A$),fc. (3.8)
The equation (3.8) implies, the existence of an su(2) valued function F = F(x, A) such that
Fk = ^-'(Jk,x^ , k = l , 2 .
9
The equation (3.5) can be integrated to get
F = A + a,where (7 is a constant sm(2) matrix.Adding term C is equivalent to a rigid motion. Hence we may take (7 = 0. The equation (3.6) is interpreted as a coordinate representation of the A family of the surfaces in su(2). The Gauss- Weingarten equations are equivalent to
and the Gauss-Mainardi-Codazzi equations {Fjki = Fjik)
(C/l.2-i72.1 + [C l,i 2]).A = 0
are identically satisfied by virtue of (3.4). □
By using the scalar product on s«(2)
< A ,B > = -\-trace{AB ) , |A| = yj< A, A > , (3.9)
induced metric gij = < F^i,Fj > = < > , i , j = 1,2 on the surface isdefined. And the frame on the surface (F i, F 2 , is determined by the normal vector
1/ =y/det{g)
3.1.3 The Fokas-Gelfand Approach
Now we will give the generalization of Sym’s formula orginally formulated in
[2].
T h eorem 3.4 [2] Let Uk — A) Ç su{2),k — 1,2 be differentiablefunctions of x^,x^ and A which satisfy (3.3) and (3.4)· Assume that the equation (3 .4 ) is independent of \. Consider the function F G su{2) implicitly given by
= A: = 1,2 (3.10)
where Ak G su{2), A; = 1,2. Then F defines a surfaces in su{2) iff the equations (3.10) are compatible i.e.
A i ,2 ~ ^2,1 + [Ai, U2] + [Ui, A2] = 0 (3-11)
is satisfied.
10
C orollary 3.5 Let us define a frame on the surface which satisfies the conditions of theorem (S.f), i.e.
, F,2 = $ - ^ 2$ , 1/ = $ -M 3$
xuhere
" U uA ,]\ ·
Then the first and second fundamental forms can be expressed explicitly as
(3.12)
(ds,r = < Ax, Ax > (dx y -|- 2 < j4 i , A2 > dx dx“+ < A2, A2 > {dx' y,
[dsiiY = < Ax,x -\- [Ax,Ux\, A3 > (dx y+ 2 < Ai ,2 + [Ai, U2], A3 > dx dx'+ < A2,2 + [A2, i/2]) A3 > (dx y.
The theorem (3.4) characterizes surfaces in terms of an arbitrary parametrization. The classical formulations of well known geometricalparametrizations can be obtain as particular cases of this theorem. An example of this theorem is given below.
T h eorem 3.6 [2] Let crj,j = 1,2,3 denote the Pauli spin matrices
0 T 0 - i 'ax = (T2 = <3
T 00 - 1.1 0 / V* 0 /
Consider an arbitrary immersion function F in implicitly defined by
F x = — , F 2 = —i^~^{bxax -|- 62( 2)^ , « 7 0, 62 7 0,
where
^,k = Uk{x\x\X)^, k = l ,2
are compatible (i.e. satisfy equation (3.3)). Then the functions
(3.13)
U u(x\x\X) = - ~ Y ,U t ( x \ x \ \ ) a „ , i = 1,2^ a=\
(3.14)
11
C/3 = i . / - ^ _ b2^dx^ dx^’ '
Ul = 1 ( 6 , £ / 2 - 6 2 C //) ,
rr3 _ 1 Vt da dbx , db2^
The first and second fundamental forms of the surface are
{dsiY = af[dx^Y -f 2ab\dx^dx' + ((¿i)^ + (62) )(cia; ) , {dsiiY = aUlidx^f + 2aUidx^dx^ + (6iC/| - b2Ul){dx^f.
are defined by
(3.15)
(3.16)
This surface is unique up to a rigid motion in space. The Gauss and mean curvatures are
K = - i f l ) + - ° ^ j ) . g = ^ wu\ - ama a a»2 a 062
A frame on this surface is given by F^\,F 2 o-nd v = —z$“ (73$.
P roo f: These results follow from theorem (3.4) with the choices Ai — —iaai,A 2 = —i{bicri + b2 <T2 ), and then the equation (3.11) become
a U ^ + 626^^ - 61C/1" = 0 , 61,1 + 6 2 t / f - a ,2 = 0 , 62,1 + a U l - = 0 .
Solving these for 11 , U2 and t/| we obtain equation (3.15). Using the results of corollary (3.5) we find equations (3.16) and (3.17). □
E xam ple 1:(Parametric Lines of Curvature) [2] Letting b\ — = 0, and introducing the notations 6 = 62, / = ^ i h = — the Gauss-Mainardi-Codazzi equations (3.4) become
d , l da . d .1 db . ,dx'^^bdx'^
dx
d , r\ 1 da
The first and second fundamental forms are
[dsiY — a [dx^Y + b [dx^Y , [dsuY = of f[dx^Y F b^h{dx^Y.
12
A frame on this surface is determined by
F i = — , F 2 = —¿$“ 6(72$ , V — —¿$“ <73$.
The Gauss and mean curvatures of this surface are
K ^ f h , H = f + h.
3.2 A Generalized Immersion Function
An important step in applying the outlined method in section (3.1.3) is to solve the following problem:
For a given differential equation in the form (3.4) with the Lax pair (3.3) find a class o f functions A\,A 2 for which one can construct explicitly the immersion function F and hence an associated surface in R^.
One of the solutions of this problem is given in section (3.1.2) via Sym (or Sym-Tafel) formula (2.3.5) and (2.3.6). A generalization of Sym’s formula was given by Fokas and Gelfand in [2]. A further generalization of this formula can be found in [3] and [6].
P rop osition 3.7 Suppose that $ is a SU{2) valued solution o f Lax equations(3.3) for a given differential equation (3.4). Let 8 he an operator representing the infinitesimal transformations. Then the equations (3.10) with
Ak = SUk + {[d,. ,8]^)<^-\ k = l,2
are compatible and F is given explicity by
F =
(3.18)
(3.19)
P roo f: The compatibility conditions can be easily verified taking into account that ($-^),)t = -^ -W k and Uk,i - Ui,k + [Uk, Ui] = 0 for A;, / = 1,2.Differentiating F we obtain the above expression for Ai, A 2 . D
All known symmetries of an integrable equation can be considered as particular cases of the 8. For instance 8 = d i is the infinitesimal generator of
13
the symmetry corresponding to translation along direction. A nontrivial example \s 6 = R where R is the recursion operator for (3.4) (if it exists).
Let us reformulate the proposition in a detailed way for the case in which the equation (3.4) reduce to a single partial differential equation.
D efin ition 3.8 The Frechet derivative of the differential function U[0] in the direction of (f>, denoted by U'{(f)), is
U’W = - U l « + e4.]l^
ruhere e is a real parameter.
T h eorem 3.9 [3] Let Uk A; = 1, 2 6e parametrized by X and the scalar function 0(u, v) where the compatibility equation (3,4) reduces to a single PDE of 9{u,,v) independent of X, Define Ak = A) G su{2) by
dUk dMAk = a
d\ ^ dx ’^[M,Uk] + U',{4>) , fc = l,2 , (3.20)
where of(A) is an arbitrary scalar function of Xj M(x^,a;^,A) 6 5г¿(2) is an arbitrary function of X the scalar (f){u,,v) is a symmetry of the equation(3 ,4 ) cLTid prime denotes Frechet differentiation . Then there exists a family of surfaces with immersions jF(a;\a;^,A) G su{2) in terms of Ai^A2 and Furthermore, F is given up to an additive constant C{X) G 5г¿(2) by
1(3.21)
P ro o f: Theorem (3.9) is a special case of lemma (3.7) where \dk,S\ = 0. It can be verified directly if Ukik = 1,2 satisfy the equation (3.3) and if is a symmetry of an integrable nonlinear PDE satisfied by then the functions Ak,k = 1,2 are defined by the equations (3.20). This implies the existence of the immersion function F.
It is possible to establish this result avoiding most of the computations. Extending the definition of a symmetry from scalar functions to functions in Lie algebras, it follows that the pair A\,A 2 is a symmetry of the pair Ui,U2 · Indeed replacing Uk by Uk + tAk for k = 1,2, the 0(e) term of the resulting
14
equation is (3.10). Thus finding a pair А^.,к = 1,2 is equivalent to finding a symmetry for the pair Uk·, к — 1,2. The pair Ai, A 2 defined by equation (3.20) corresponds to three different types of symmetries:
(i) The integrable equation (3.4) is independent of A, thus A translation is a trivial symmetry for this equation. This yields ,
Ak = > к = 1,2.
where a = «(A) is an arbitrary function of A.
(ii) Equation (3.4) is invariant under the gauge transformation Ф —»■ 5Фand
dSU k - ^ S U k S - ^ ± ^ S ~ ^ , к = \,2. (3.22)
Letting S = I + tM, where / denotes the identity matrix, the expression in (3.22) become,
Uk Uk T [-^> 4" 1 k = i ,2.
Thus
Ak = ^ + \M,Uk] , k = l,2.
(iii) Let </> be a symmetry of the equation of (3.4). Then Frechet differentiation gives
Uk ^ Uk{0 -h e<l>) = Uk + cU'k{4>) + 0 { e % ¿ = 1,2
which implies:
Ak^U'k{cj>), ¿ = 1,2.
Linear combination of the above three symmetries gives rise to equations (3.20).□
E xam ple 2: (Theorem (1.2) in [2]) Let
M = /i(x\x^)i/i + f 2{x^,x^)U2 + Mo
where Mq € su(2) is a constant matrix and a(A), / 2(3:^ a: ) are scalarfunctions with the arguments indicated. Then equations (2.3.20) become
du, , dh , dU, , df2dx dx^
U2
15
T
+ / 2 ^ + /3|M„,c/,j + (/;(^ ).
J - ,.ÎU 3. n 3. f 3. n ^-42 - “W â r + â? 5?· + a? +
^ T T
f 2g ^ + f3[Mo,U,] + Ui(^),
and immersion function takes the form1 / \
F = $ - M a — + / i d l $ + /2^,2$ + Mo$ + ^\4>) j .
E xam ple 3: (Parallel Surfaces) if F in su(2) is parallel to F then F — F = au = where i/ is the unit normal vector (i.e. < A^,Az > = 1) to thesurface F (also to F ) and a is a constant (distance between surfaces). One can easily observe that parallel surfaces can be given by virtue of the generalized immersion function. It is enough to set M and <f> — and
F = $ , au = a$
R em ark 3.10 In sections (3.1) and (3.2) we have considered surfaces in as surfaces in su(2) algebra. The whole approach for immersions of dimension dim M > 2 becomes considerably more difficult. But the notion given in proposition (3.7) can be extended to immersions into a lie algebra g (let dim g = m) of higher codimension. Let
^,k = U k^, k = l , - - - , n < m (3.23)
denote the system of equations where Uk{x, A) are smooth functions of A and coordinates x = (x^, ■ · · , a:” ). The functions $ take values in a semisimple matrix group G and Uk € 9, the lie algebra of G. The integrability conditions of this overdetermined system of equations require that
Uk,i -Ui^kF[UkM = . k < l = l, ,n (3.24)
Equations (3.23) can be interpreted as defining a G-valued connection (G representation of Gauss-Weingarten equations (A.f) and (A .5)) with equation (3.2f) (Gauss-Mainardi-Codazzi-Ricci equations in this representation).
We now need a prescription for constructing an immersion F associted with the symmetry of (3.2f) ■ Introduce an arbitrary variation 6. The from (3.23) we get
(i$)_fc = 6Uk^ + UkS^
16
and consequently
{^-^S^),k = ^-\6U k + [dk,6])^
so that there exists a function F : R ” g such that
F,k = ^-^6Uk^
and
F = ^-^6^ + C, (7 eg.
Notice that C = (7(A) ¿5 arbitrary in this last equation. For this matrix group G we calculate the geometrical quantities by using nondegenerate invariant bilinear form g X g ^ IR
9ki = < > = < SUk, SUi >, / = 1, · · ·, n
and by introducing normal vector fields i/rj r = n + l , - -* ,m = dim G
Pt = < -UiSUk + SUk,i + SUiUk, i r >
This formulation gives whole algorithm for constructing the soliton immersions:
(i) Find a soliton system with Lax represenntation (3.23) for which n < dim g.
(a) Construct an orthonormal basis for g.
(Hi) Construct a function F : —> g from a variation 8 : Gdefines a canonical map (7 ^ g under left translation.
TG which
This approach is similar to the one developed by Sym (Sym considered only the immersed submanifold with dimension 2), and the recent work of Dodd gives this construction for arbitrary dimension and codimensionj [5].
3.3 Immersions Associated With The Symmetries of The Integrable Gauss-Mainardi- Codazzi Equations
The theorem (3.9) provides an algorithmic approach to construct the surface by starting from a suitable Lax pair. We shall apply this technique to construct
17
surfaces associated with the constant gauge transformation for arbitrary Lax equations and associated with symmetries of the Sine-Gordon equation
3.3.1 Immersions Associated With the Constant Gauge Transformations
Now let us work out the surfaces generated by the constant SU(2) rotations of i.e. by a constant su(2) matrix Mq
T h eorem 3.11 [1] Let Ak = [Mo,Uk]-,k = 1,2 , where Mo 6 su(2) is a constant matrix. Then K = H = , where e — ±1 and\Mo\ — \/< Mq., M o > . Hence all such deformed surfaces are spheres with radii I Mo I where the immersion function is
F = Mo
P roo f: Let Uk = \Y/a=\^k for A: = 1,2 be any Lax pair and Mq = aa be a constant su{2) matrix, where cr„, j = 1,2,3 denotes Pauli
spin matrices. Since [(T„, crp] = 2itoi/3.y(T.y, we ha.ve
Ak = [Mo, Uk] = -^£«/37 ^7·
To calculate the normal vector field v = we need [Ai,A2]
[yll, yd'j] = 2 £a/?7 a6C £f;i/3 ^
= ( /0fi 7C - 0C 75) rn u^ C/| a.y- t M q
since
and
C/f rn U2 (Tc = (< U2,[Mo,M o] > ) Ui = 0
— — - 4 (< Mo,[U2 ,Ui] > ) Mo.
Letting £ = T , we find
18
(3.25)
(3.26)
< - 1,1) - 3 > = < - 3 > = < - 2,2) As > = 0.
Using these equations it follows that
e
hence
|A/ol 9ij ·
Hence S = / , where / is the identity matrix. Thus
/lT = dei(^) =1
H = tr{S) = -
|Mor2e
IMol’
claim follows. □
(3.27)
(3.28)
(3.29)
(3.30)
This result is quite interesting. La,x pair is arbitrary so that under the rigid SU{2) rotations all integrable equations are mapped into a sphere.
3.3.2 Immersions Associated With The Sine-Gordon Equation
Both in the classical differential geometry and integrable nonlinear partial differential equations, sine-Gordon equation for smooth function 0{x^,x'^)
d H- sin 9, (3.31)
dx^dx'^
is of special interest. The Gauss-Mainardi-Codazzi system of any pseudospher- ical surface endowed with the so-called asymptotic coordinates reduces to the sine-Gordon equation.
The Lax pair for the sine-Gordon equation is given by (3.3) with
^1 = ^ (-^,1 <3) , U2 - - cos Oas), (3.32)
where 9{x^,x^) G M and A is an arbitrary constant. Let be a symmetry of equation (3.31), i.e. let y? be a solution of
aydx^dx'^
= if cos 0. (3.33)
19
There exists infinitely many explicit solutions of equation (3.33) in terms of 6 and its higher derivatives. The first few are (see [26])
01 6%^,1 ) ^,2 , ^,111 + , 0,222 + , (3.34)
starting from the third one all such solutions are called the generalized symmetries of (3.31). Then for each ip theorem (3.9) (with a = 0, M — Q) implies a surface constructed by
i dp^ ^p(cos0(T2-l·sinOaз). (3.35)
We now study the surfaces corresponding to these generalized symmetries.
L em m a 3.12 fSj Let M be the surface generated by a generalized symmetry of the sine-Gordon equation. That is, let M he the surface generated by Ukand Ak,k = 1,2 defined by equations (3.32) and (3.35) respectively. The first and second fundamental forms, the Gaussian and the mean curvatures of this surface
are given by
F = $-1
ds]i = ^{Xp^\sii\9{dx^y'+ ^pd^2[dx^Y),
4A^0,2sin0 2A(c/?,i 0,2 + <|i’ sin^)K = --------------- , H = ----------------------------.
W .i
(3.36)
(3.37)
(3.38)
P roo f: Applying corollary (3.5) to the frame defined by (3.35) we get
and
9n =
9\2 =
922 =
6ll = < 4li,i
b\2 = < All,2
i>22 - < A2,2
< Ai,Ai > =4 ’
)£L4A2’
Xp i sin<?2
t 0,22A
20
where
A3 = —i (sin 9(72 — COS 6 (7 3 )
Using the equation (3.37) the Gauss and mean curvatures (3.38) are obtained directly. □
An immediate corollary of the above lemma is:
C orollary 3.13 [1] Let M he the particular surface defined in the above lemma corresponding to (p — 9 2- Then this surface is the sphere with
ds] = l(s\n^e(dx'Y + ^^{dx ‘ n
K = 4X' , 7/ = 4A. (3.39)
We now present a global result regarding the above surfaces.
T h eorem 3.14 [1] Let M be the surface defined in lemma (3.12) in terms of a generalized symmetry of the sine-Gordon equation. If M is a compact, connected and oriented surface then it is horneomorphic to a sphere.
P roo f: All compact, connected and oriented surfaces with the same Euler-Poincare characteristics are homeomorphic, [34]. For compact surfaces the Euler-Poincare characteristics y is given by Gauss-Bonnet theorem
X = ^ / ^ det{g) K dx dx' .
Since det{g) = then the integrand yjdet{g) K simply becomes
\Jdet{g) K = A 9 2 sin 9.
Hence X is independent of symmetry <p
AX = — f i 9 2 s\n 9 dx dx .
27f 1 J s
(3.40)
(3.41)
(3.42)
21
This proves that x has the same value for all generalized symmetries and hence for all sine-Gordon deformed surfaces. Thus in order to calculate x it is enough to choose the simplest case. According to the Corollary (3.13) the choice ip — 9 2 leads to a sphere with radius ^ where x = 2. In this example since curvature density have the same form with eqn. (3.41), then x > 0 for all compact sine-Gordon symmetry surfaces. Hence (with orientation) all deformed surfaces have the Euler-Poincare characteristics x = 2. Therefore they are all homeomorphic to a sphere. This completes the proof of the theorem. □
R em ark 3.15 Consider the case, immersion (3.36) is smooth, in the preceding theorem. Since continuous and smooth categories are same for two dimensional manifolds, then compact, connected and oriented surfaces associated with the symmetries of sine-Gordon equation are diffeomorphic to sphere. If there are any such surfaces other than the sphere with K > 0 then they must be ovaloids.
Solitonic solutions of the sine-Gordon equation satisfy the rapidly decaying conditions , ^(±oo) = 0, 0 ,i(±oo) = 0, ^ 2(±oo ) = 0,.... Then for such a case we have the following lemma
L em m a 3.16 [1] Let M be the surface defined in Lemma (3.12). Suppose that this surface is non-compact. If the associated solution 9[x^,x^) of the sine-Gordon equation satisfies the conditions that 0,9^\,9^2i··· tend to zero as x ± oo then
/ OO P O O ! -------------------
/ Jdet{g) K dxUx"^ = 0. (3.43)-OO J — oo
P roo f: ( ,2) |ci tends to zero as Ci,C2 ±oo so (3.43) does.Q
We now consider a different class of immersed surfaces which are also constructed from solutions of the sine-Gordon equation such as
^ 51 ·
L em m a 3.17 [1] Let M be the surface constructed by Uk,k = 1,2 which are given by the equation (3.32) and by Ak = = 1,2 гı)here f.i depends on A.Then M is a surface of constant negative curvature.
22
P roof: Frame on M is defined by
^1 = 2^^^’i jJL
A 2 = (sin (72 - cos CTa),A3 = ± i CTi. (3.44)
Corollary (3.5) allows us to calculate the geometrical quantities given above. This surface has the following fundamental forms and curvatures
^ cos 6 dx d x ^ [d x ^ Y ')^dsjj = ± ‘ s'mO dx dx^,
K = - ^ , H = ± — cot(0).
□
C orollary 3.18 [1] Let 9 be a rapidly decaying solution of the sine-Gordon equation and M be the surface defined in lemma (3A7). Then
/ 00 roo !---------/ Jdet{g) K dx dx = 0.
-oo J — oo
P ro o f: This is a consequence of
\Jdet{g) K = — sin = — ,12·
□
We now consider yet different class of immersions associated with solutions of the sine-Gordon equation, in the form
F = (m 5a + ? <Ti) $
L em m a 3.19 [1] Let M be the surface constructed by Uk,k = 1,2 which are defined by equation (3.32) and, by Ak = k = 1,2 with ¡j, = Xp.Then
ds] = — (A (dx^Y — 2 sin 6 dx dx 4- — (dx^Y),2 A
ds] · = - [A {dx^Y — 2(sin 9 cos 9) dx dx -f — [dx^Y],2 A
23
2 2 2K = — - tan 6 , H = -------- tan 9.
P‘‘ P P
The curvature density yjdet{g) K has a form similar to the one in corollary (3.18). Thus yjdet(g) K = — sin $ -- —i?,12-
P ro of: Frame on M defined by
1P {( 2 +(rz)·,
% T)A 2 = - Y ((cos^ + sin0) 0-2 + (cos 0 — sin0)<T3),
Z AA3 = ¿(71.
Then claim follows by using corollary (3.5).□
The following corollary of the Lemma (3.19) is for the solitonic solutions of the sine-Gordon equation
C oro llary 3.20 [1] Let 9 be a rapidly decaying solution o f the sine-Gordon equation and M be the surface defined in lemma (3.19). Then
/ 00 roo J--------/ Jdet[g) K dx^dx^ = 0.
-00 J —00(3.45)
24
Chapter 4
Weingarten Surfaces
In this chapter, making use of the generalized immersion function established in chapter 3, we shall construct Weingarten surfaces arising from some other nonlinear partial differential equations. The classical description of Weingarten surfaces is studied in [35].
4.1 Linear Weingarten Surfaces
In this section we will study equations on which the surfaces associated with their symmetries hold a relation
f { K , H ) = a K + ß H + j = 0.
4.1.1 The Sine-Gordon Equation
Now start from the Lax representation of sine-Gordon equation given in equation (3.32):
z z= 2 + < 3) , U2 = — {sïnO(T2 - cosOas).
L em m a 4.1 [1] Let M he the surface constructed from Ui and U2 defined by equations (3.32) and from Ak = = 1)2. This surface whoseimmersion function is given as
25
satisfies the following Weingarten relation
(li'‘ + K + 2pX'‘ H + = a. (4.1)
and it is parallel to a space o f negative constant curvature. The distance between these surfaces is
P roo f: Weingarten relation (4.1) can be directly verified with a tedious calculation.
Let Ko and Ho be the Gaussian and mean curvatures of a surface Mq with constant curvature K q and let M be parallel to Mq then
Ko =K
, Ho =H - a K
(4.2)l ~ 2 a H + a ^ K ’ " l - 2 a H + a^K
where a is a constant [34]. Hence comparing the first equation above and (4.1) we find that
_ P _ 16 A"4 ’ ” 3p’ + 4/j2'
Thus M is parallel to a surface Mo with negative constant curvature and | is the distance between the surfaces.□
We have the following corollary to the lemma (4.1).
C orollary 4.2 The surfaces equidistant to pseudospherical surfaces are linear Weingaten surfaces and according to lemma (4-1) in a certain coordinate system all such surfaces can be characterized by the sine-Gordon equation.
4.1.2 The Sinh-Gordon Equation
The sinh-Gordon equation defined by
10,U + 0,22 + ¡ { n y ^ - = 0 (4.3)
where 0{x^,x'^) € K and Ho ^ 0 is real constant. This equation usually associated with surfaces of the constant mean curvature Ho- In what follows we will show that this equation can also be used to construct several other classes of interesting surfaces.
26
L em m a 4.3 Let${x^,x^) be a solution of the sinh-Gordon equation (4-3), where is a real constant. Define Uk,Ak € su{2), k = 1 , 2 , by
U\ = jfcos A(//oe® + e ®)<Ti — sin \{Hoe^ - e + 20,2( 3],U2 = —^[sin A(//oe® + e ^)cti + cos A(jÎ/qc — c )< 2 "I“ 20,1(73],
At = + Yl<’ 3,Uk] , k = l,2
(4.4)
(4.5)
where ¡i and p are real constants. Then the associated surface M xuith the immersion function
F = $-1 {2fi 5a + ^ <Ts) $
satisfies the following Weingarten relation
(p 2 _ 4 /i2 ) /i + 2 p iï + 4 = 0. (4.6)
There are some particular limiting cases, If p = =t2/i S is a surface of constant mean curvature
_ _ r), £7 _ 1 Hq—1p — 2 fj. , H — —- , A — 42 //| giff,
p = -2 /i , , K = - ’- ^ .
I fp = 0 , S is a surface of constant positive Gaussian curvature,
K = ^ ,TI ( 2 \ + l
If p = 0 , S is sphere.
P roo f: Direct application of the theorem (3.9) and corollary (3.5) gives the stated result. The surface M associated with the symmetry given in (4.5) has the following fundamental forms and curvatures
19n =
012 —
022 —
j g ^ „ ( № " ( 2 p + p ) + ( p - 2 / . ) r +
4ifo (4/1 — p ) sin A e ®),Ho {4p^ — p ) sin 2A
8 ’
{[e' ‘’Hoi2p + p ) - { p - 2p)Ÿ -1
16e24 Ho (4/1 — p ) sin A e ®)
27
bn
612
622
—Hq {p + 2 p) — p + 2 p — 2 pHo cos 2A
pHo sin2A8e 20
-Hq (p + 2 p) — p + 2p + 2p Ho cos 2A e'·' 8^20
20
K = 4 e^^H - 1
H = - 4
e^^H^{2p + p y - { 2 p - p y ^e^^H^{2p+p) + { 2 p - p )
e^^Hi{2p + p y - { 2 p - p y and satisfies the following Weingarten relation given in (4.6). By arguments similar to the ones used in proving lemma (4.3), it can be shown that this surface is parallel to surface whose curvature is
16Ko = 16p - 3p2
constant. Distance between surfaces are a = f □4
C orollary 4.4 The surfaces equidistant to constant curvature surfaces are linear Weingaten surfaces and according to lemma (4-3), in a certain coordinate system all such surfaces can be characterized by sinh-Gordon equation.
In the case oi Ho = 0 the sinh-Gordon equation has some particular geometric interpretation. The sinh-Gordon equation reduces to the Liouville equation
O,u-l·0,22-\e-^‘ = O■ (4.7)We have the following lemma:
L em m a 4.5 Let 0 [x^.,x^) G IR 6e a solution of the Liouville equation (4·' )· Define Uk, Ak, k = 1 , 2 by
U\ = - ( e "cos A cTj-fe sin A (T2 -|-20 2 <3),
U2 = —-(e~® sin A (Ti — e“ *’ cos A <T2 -|-20,1 (T3),
where Ak,k = 1 , 2 are given in (4-5) with p ^ ±2p. The associated surface S has the following fundamental forms and curvatures
4K =
H =
(2/2 - p)2 ’4
2/2 - p ·
28
Thus for any fx,p with p ^ 2p, S is a sphere.
P ro o f: Direct result of theorems (3.9),(3.4) and corollary (3.5) with symmetry given in (4.5)).D
4.2 Nonlinear Weingarten Surfaces
4.2.1 The Nonlinear Schrödinger Equation
The nonlinear Schrödinger equation is an equation for a complex function ^(a;\a;2).
¿ >,2 = +2|V>|^.
Letting \j){u^v) = r{u^v) + is{u^v)^ the real valued functions r and s satisfy
(4.8)r,2 = 3,11 + 2 s ( r 2 + S ^ ) ,
3,2 = - r , i i - 2 r ( r ^ +3^).
The associated matrices Uk^k = 1,2 defining NLS’s Lax pair are given by
—2A 2(3 — ir) \2(3 + ir) 2\ J ’= i l .
U2 = - I-4A + 2(r^ + 3 ) Vi — iv2 \
vi + iv2 4A — 2(r^ + 3 ) / ’
(4.9)
wherevi = 2r,i + 4A3 , V2 = —23,1 + 4Ar. (4.10)
L em m a 4.6 Let Uk,k — 1,2 6e defined by equations (4-9), where r and s satisfy the integrable nonlinear equations defined by (4-8), and i>i,V2 be defined by (4-Í0). Let Ak be defined by Ak = n A: = 1,2, where y, is a real constant, i.e. let
i Í - 2 y 0 \ _ i f -8A/Í A y { s - i r )^ ^ 2 ^ 0 2//J ’ ~ 2 \ A y { s + ir) 8A/i
(4.11)
Then geometrical quantities of the surface M with the immersion function
^ T\
29
associated with the Uk,Akik = 1,2 are
ds] = [{dx - d x ^ y ^ {dx^)\dsji = -2/j.q[dx^ - (-^_i + 2X)dx^y + 2iJ.q^uidx'^Y,
K = Q.nH^q
_ 9,11 ~ 9 (^,1 + 2A^) — Aq^2 ixq
which can be expressed in terms of the new variables
r — q cos ^ , 5 = sin <,
In terms of these variables the NLS (4-8) become
(¡<f>,2 = -q ,\ - i - 2 q^-[■ q<f)\,9,2 = 9?^,ll + 29,1 </·,!.
(4.12)
P roo f: Use frame defined by (4.11) and corollary (3.5).□
In particular if ^ = ux" , q = q{x^) , where 1/ is a real constant, then q{x^) satisfies
q " = - 2 q ^ -i,q . (4.13)
L em m a 4.7 Let Uk,Ak,k = 1,2 be defined by the equations (4-9), and (4-il) where r = q'(a; ) sin(i .x ), s = q(x^) cos{i'Tfi)f A,i/,// are constants and q{x^) satisfies (4-13). Then the associated surface S is a Weingarten surface which satisfies the relation
2 p fH ‘ [jjf K - v ) = (3/i A" + 4 - 2 v f .
If u = —4A the above Weingarten relation becomes quadratic,
r. 2 ,,2 4A2 ^
4.2.2 The mKdV Equation
Let p[x^,x^) satisfy the so called modified Korteweg-de Vries equation
3 2p,2 — p,\n + 2 Pp-
30
(4.14)
The associated matrices£/fc, = 15 2 which define its Lax pair are given by
A -p \
where
= U W ’\ -p
U2 =Vl+iV2 ^ ) ’
(4.15)
\2 ^" h — P,i\ + - — A p , i>2 — —Xp,i· (4.16)
Lem m a 4.8 Let Uk,k = 1,2 be defined by the equations (4 .15), where p[x^,x^) G K satisfies the mKdV equation (4-14) o-nd Vi,V2 be defined by the equation (4-16). Let Ak,k = 1 , 2 be defined by Ak = p ^ ^ ,k = 1,2, where p is a real constant, i.e. let
= i
A 2 - i
P 0 A 0 - / / / ’
+ 3/xA -2pXp + ipp i(4.17)
—2 pXp — ippp ---- 3pX J
The geometrical quantities of the surface M associated with the Uk, Ak, k = 1 , 2 ,
F = pdX
are given by
K (p‘\+A\·^ [4p^^,iiii -^P ^ P,I P,in -4p^(p,ii)^+4/> P i p,n - 4A2 + 4p5 _ 4 ^ 3^4
H = 2j + 4A 2/ 9 2 ) 3 / 2 [-p p ,\n i + P,1 p,n\ - ‘ X^PPm
-P^ p ,ll + 2A2 _ 3^2 ^2 _ 4^4 p2 _ 4J^2 ^4j^
(4.18)ß(p
dsj =dsh
^ l(dx + l ( p ^ - 6A^) dx fi + (p , + 4A^ P ) (dxyj,,2+4A2p2)l/2 l-P^ {dx^y + ( - 2 p p , n + P4 + 2A^ p2 _ ^4) ¿J.ldx2(p
"hi (~4pp^iiii + 4p i p,ini + 12A P P,n — 8 p P ,u ~ 4A P4 — ap p i — 4A + 4A p‘ — p ) {dx^Y
(4.19)
P roo f: Direct result of equation (4.17) and corollary (3.4). □
A particular reduction of the above surface M is a Weingarten surface with a complicated Weingarten relation.
31
C orollary 4.9 LetUk^k = 1,2 6e defined by the equations (4-15), where a are constants and suppose as a particular case, that p[x^,x^) = p[x + ocx ) satisfies
p" = a p - ^ . (4.20)
Then the associated surface M is a Weingarten surface satisfying the relation
^2 i/2 p2 + 4A2) _ p2]3 ^ 16A2 [ / -6/>2 (a + 4A2)-8A2 (a + 4A2)]2,
where
16 A2
(4.21)
— 4(a + 4A^) +a + 4A
It is interesting that using a different Lax pair for equation (4.20) it is possible to obtain a Weingarten surface simpler than the above one in (4.21)
L em m a 4.10 Let Uk,k = 1,2 defined by
A -p A- p - a ; ’
C/i = f
U2 =^ - ( a + aA + A ) (a + A )p - ip ,i
(of + A) p + zpp — Y + (q! + «A + A ) /
where A, or are constants and p satisfy the equation (4-20). Let Ak,k defined by Ak =
' p 0 A
(4.22)
1,2 be
A, = f
A 2 —
0 - p j
f —{ap + 2 pX) pp“ \ p p ap-\-2 p\
This surface M with the immersion function
.1dX
F = p
is a Weingarten surface satisfying the relation
2^2 {p K + 4a) = [3p K + 4A" + 8«]^
In the special case a = X the relation becomes
2p H' = 9[p^/A + 4A ].
32
(4.23)
(4.24)
(4.25)
(4.26)
P roo f: The geometrical quantities of the surface M associated with the Uki A: = 1,2 are given by
K = ^ [p -^ -2 a \ , i 7 = ± [ 3 p 2 + 2 (A 2 -« ) ] ,
ds] = !^[{dx^+ {a + 2 \)d x '^ f^dsh = ^ [dx^ + {a + X)dx^Y + f ( p ^ - 2 a ) { d x y .
by using corollary (3.5).□
33
Chapter 5
Conclusion
In this work we presented a procedure of the construction of surfaces in associated with the symmetries of integrable nonlinear partial differential equations within the framework of surfaces on Lie groups and on Lie algebras. We applied this method to some well-known integrable equations and obtained several symmetry surfaces. In particular we investigated some global properties of surfaces arisen from constant gauge transformation and symmetries of sine-Gordon equation. We showed that under rigid SU{ 2 ) rotations all integrable equations are mapped to sphere. In the case of sine-Gordon equation, we proved that all compact sine-Gordon symmetry surfaces are homeomorphic to sphere. Besides we have constructed several Weingarten surfaces associated with symmetries of some soliton equations. We found some explicit linear and nonlinear Weingarten surfaces generated by the symmetries of sine-Gordon, sinh-Gordon, mKdV and nonlinear Schrödinger equations. Some characterization results are given for linear Weingarten surfaces.
However, many questions remain open and deserve further investigation. The logical continuation seems to consider the equations whose Lax pair is given on other Lie algebras e.g. su(l, 1 ),5/2(1 )· Another interesting problem is the characterization of nonlinear Weingarten surfaces by the symmetries of soliton equations. These questions are being pursued further.
34
Appendix A
Fundamental Equations for Submanifolds
In this appendix we begin by considering higher dimensional embedded or immersed manifolds, of higher codimensions. For interested readers we refer to [32, 33].
A .l Fundamental Equations for Submanifolds
Let F : N' be an immersion of an m-dimensional Riemannian manifold(M, F*g^) into an n-dimensional Riemannian manifold [N,g^). For every p 6 M, we have TpN — TpM © TpM^^ and we use this decomposition to define two projections, T : TpN TpM and ± : TpN TpM^. V{TM) and r(TM·*·) are the sets of tangent and normal vector fields respectively. For vector fields X , F e F(TM ) and i e F(TM ^) we write
V J F = T ( V « r ) + ± ( V " r ) .V K = T (V K ) + X (V K ).
(A .l)
where V y denotes the connection in N. and X V y induces connectionsX
on T M and on T , denoted as and D x respectively. And we will denoteA ^ X ) = - T ( V K ) .
D efin ition A . l The second fundamental form tensor of M is s(A , F ) = i (v !? v · ) . / / we choose i^m+i, ■ ■ ■, € V{TM^) such that < > — rswhere trs = defined in a neighborhood of a point p G U C M , we definen — m real valued second fundamental forms fi’' by
Y y = < V^F, Ur > = < s{X, F), Ur > .
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(A.2)
D efinition A .2 Connection D defined above is called the normal connection. So we introduce the normal fundamental forms ¡5 , by
ßr{X) = < > ■ (A.3)
With notation that we have just introduced , we may rewrite decompositions given in equations (A .l)
V j y = T (V ? K )+ s (X ,K ) ,
V Î Î = A, {X ) + D x i,
(A.4)
(A.5)
which are called the Gauss formula and the Weingarten equations respectively.
T heorem A .3 Let M'^ he a submanifold of the Riemannian manifold N' , for A , Z and W are tangent fields along M , we have the Gauss equation
(A.6)< R ^{X , Y)Z, W > - < R ^ { X , Y)Z, W >= < s{X, Z ), s{Y, W ) > - < s{X, W) ,s{Y, Z) > .
P roo f: We have
V ^ V ^ Z = V ^ V ^ Z + s(X , X ^ Z ) + V ^ (s (r , Z )),
similarly
V ^ V ^ Z = V ^ V ^ Z + s (r , V ^ Z ) + Vi^(s(A, Z )),
as well as
^ix,Y)Z = '7^x,ytZ + s ( [X,Y] ,Z) ·
Substituting last three equations and noting that W is orthogonal to any term .s(.,.), we obtain
< R^{X, y )Z , W > = < R ^ { X , Y)Z, W >+ < V ^ (s (r , Z )) - V(y(s(A, Z )), W > .
On the other hand, since < s(F, Z ), W > = 0 we have
0 = A < s{Y, Z ), W > = < V ^ s (r , Z ), W > + < s(r , Z ), s(X, W) > .
Desired result is obtained by substituting the last equation and the similar expression with X and Y interchanged. □
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T h eorem A .4 Let M be a submanifold of a Riemannian manifold N. Let Ur G T(TM·^) where r = m + 1, , . ,n with corresponding fi’' and 01. Then for all tangent fields X , Y, Z along M , we have Mainardi-Codazzi equations
(A.7)< R ^ { X , Y ) Z , Ur > = i X ^ n ^ ) { Y , Z ) - ( V ^ n ^ ) { X , Z )
+ trt{a^{Y, Z)01{X) - W { X , Z)01{W)).
P roo f: Since D is the connection on the normal bundle of M we get
Dx{s{Y, Z) = X{Ü^{Y, Z))u, + n “Dxu,,
moreover
s (V ^ r , Z) + s(Y, Z) = r , Z)u, + i2 * (rv ^ , Z)u,.
Then these two equation give
Dx( s ( Y, Z) - s ( V ^ Y , Z ) - s ( Y X ^ Z )= ( v ‘ a -)(Y ,z ) + w (Y ,z )D x r.,
and hence
< D x ( s ( Y , Z ) - s ( V ^ Y , Z ) - s ( Y , V ^ Z ) , u . >= (v y s î ') (y , Z) + t.,(n '{Y , Z)!3‘,(X ).
Finally there is a similar equation by interchanging X and Y . After the substitution Y R ‘ {X, Y ) Z = Dx{s{Y, Z ) - s ( V f Y , Z ) - s { Y , V ^ Z ) - {Dy (s{X, Z) - s i X y X, Z) — s(X, V y Z)), we obtain the Mainardi-Codazzi equations. □
Before introducing the Ricci equations, we introduce one more operation. Given tangent fields X and Y along M and a basis Ui,..,Um such that < Ui, Uj > = g = ±Sij, we set
O’- * Y) = g^^Ü^{X, Ui)ü‘’{Y, Uj).
T h eorem A .5 Let M be a submanifold of N. If Ur Ç: r(TM·*·) xuhere r — m-|-1, ..,n with corresponding ÎÎ’’ and 0 , then for all X ,Y Ç. F(TM ), we have the Ricci equations
< R ^ { X , Y ) u r , u , > = n ^ * ü ^ { X , Y ) - r t ^ * ü ‘>{Y,X) . .+(vi^ /3n (K ) - ( v " / j ; ) ( j c ) + ,,x {f);(x )is i(Y ) - K{ Y) i i ; (X) ) . ' ' ’
P roo f: By using the Gauss formula and Weingarten equations
= - ^ x M y ) - + Dx Dy C,
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we get the normal components
J - i V i V f i i ) = -s(X, A^Y)) + DxDyi,
J - (V ? V R ) = -ir(F ,4e(A :)) + D y D K i ,
and
Thus we obtain
LR'^(X, Y )( = Rd (X, Y )( + i(A{(X), Y) - s(A ((Y ),X ),
where Rp denotes the curvature of the normal connection. Now if U i , U m is a given basis of TpM
and
we also have
A,^{X) = n{X,Ui)Ui,
< s{A, (X), V), 1/, > = * n ^ ( X , V ) ,
< D xD yu„y. > = X(ßt(Y)-e,Hßl(X)ß‘K(Y),
< D[x,y,K, y. > = ß tiV ^ Y ) - ß ; { v ^ x ) .
Using the above equations we obtain the Ricci equations. □
We have seen that the Gauss and Mainardi-Codazzi equations are precisely the integrability conditions of the Gauss formula.The integrability conditions of the Weingarten equations lead to two sets of equations. One set reduces to the Mainardi-Codazzi equations, the other set is the Ricci equations. Therefore three fundamental equations give the complete set of equations for smoothimmersions.
A .2 Gauss, Mainardi-Codazzi and Ricci Equations in Local Coordinates
We shall rewrite Gauss, Mainardi-Codazzi and Ricci equations in terms of a basis [32] :
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Let M be an immersed submanifold in N. We consider a coordinate system j/\ j/” on a neighborhood of U C N, with the metric
< ; - > ” =giedy-<8,dy'‘ ,
and let be the coordinates on M, with
< ·, ■ > ^ = gfjdx^ ® dx\
When we consider the local parametrization of M a,s y°‘ = y°‘ {x^, ..,x'^). di for i = is the corresponding basis of TpM.
9cupV'.iV·.] 9ij )
where i/“ denotes the covariant differentiation of y°‘ with respect to x\
To write the Gauss, Mainardi-Codazzi and Ricci equations, all we need is to write fi’’ and /?* in terms of the local coordinates. For the local parametrization y" = x^) of M, we can write the basis of TpM in terms of coordinatesof N as di = y^do, ■ Then we have
= (!/S + r ; , ! / ' ! / ' ) « . .
and definition of O’· gives
nr. = n^(di,dj) =<V^.dj,i^r > (A.9)
Now we have
VS;.', = + r-„v>,y2d...
Similarly normal fundamental forms can be written in terms of local coordinates
= p^{di) =<V^.Ur,Us>
= gape's i<.i + p<T ?y:i) c,·(A.IO)
Now we can write all fundamental equations of immersed manifolds Let X = di = yfida, Y = dj = ryfjdp, Z = dk = y'Jkd , and W = di = y?{ds, then we have the following lemma
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Lem m a A .6 In local coordinates the Gauss-Mainardi-Codazzi-Ricci equations are respectively given by
= -R." 1 + £ . . ( « « « ' , - o j i l i j , (A. l l )
and if Vr = and we get Mainardi-Codazzi and Ricci equationsas follows
- K K i h (A.12)
p;,., - 0 k j + £.»W*/?i,· - A 'Æ )·(A.13)
A.3 Immersions into Constcmt Curvature Spaces
It is useful to examine the form which our fundamental equations take when the ambient space N has constant curvature K q. For the case K q > 0 the manifold N is just the n-sphere of radius . For A' o < 0 , we obtain an analogous submanifold n + 1 dimensional Euclidean space A"·*·’ by considering a pseudo-Riemannian metric on K q = 0 is trivially A".
In each case mentioned above, curvature tensor of N satisfies
< R ''(a „d e}d ^ ,a , > =
where do,, dp, d-y, ds are the basis element of TpN
Now Gauss, Mainardi-Codazzi and Ricci equations can be simplified by using this restriction on R!^. They take the form
the Gauss equations
K g { £ ,g l - g^ygSs)v°y^yhi‘, = - n ',“ ; , )„ N „.6 M
the Mainardi-Codazzi equations
A 'o is iiii ; - g ^ M y r d v l ^ = n,'fci -
the Ricci equations
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respectively.
As a final remark we can write fundamental equations of immersion of M into N = E' i.e. curvature tensor — 0, take the form
the Gauss equations
- m u ) = 0-
the Mainardi-Codazzi equations
(i’.ki - nii:, + - n ’t/r.i) = 0.
the Ricci equations
+ ««(A*/?.* - fitA ") = o.
respectively.
A .4 Immersions of Hypersurfaces
Let us consider a more specific situation where M'^ is a hypesurface in ,that is, a submanifold of codimension 1. In the case of hypersurfaces we can locally choose a unit normal vector field on a neighborhood of p G f / C M. Then Weingarten equations reduce to
< Y > = - < i y , > = < i/, s{X, Y) >,
and since < i/,Y > = 0 and < u,u > = I along M,we have
0 = X < i / , r > = < V ^ i / , y > + < i / , V ^ r > ,
0 = X < u,i> > = 2 < V^i/, 1/ > .
Hence
Dxu = 0,
s ( A , T) = fi(A ,r )z /,
or equivalently normal fundamental form ßnX\ vanishes. We can compute
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the Gauss equations
and the Mainardi-Codazzi equations
The Ricci equations are trivial if M is a hypersurface. And finally if we let N have zero constant curvature, then equations simply become
the Gauss equation
- m ] k ) = 0.
and the Mainardi-Codazzi equations
respectively.
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